Calculus I · 2B · lesson
The First Derivative Test
Learn the first derivative test with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Theorems, extrema, and curve shapeWhat this section is building
Learn the first derivative test with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Critical numbers divide the domain into testable intervals; endpoints and discontinuities keep local evidence from becoming an unjustified global claim.
List the domain and candidates, test derivative signs, compare endpoint values, and verify each theorem's hypotheses explicitly.
Calling every point with f-prime zero an extremum or every point with f-double-prime zero an inflection point.
Learning objectives
Use changes in the sign of to classify local extrema.
Classify Local Extrema by Sign Changes
Before the formulas
Graph analysis in The First Derivative Test is a coherent reconstruction problem. Domain, intercepts, limits, derivative signs, critical points, concavity, and asymptotes constrain the same curve. Build the picture in layers rather than trying to sketch from the original formula at once.
Keep a feature table. Each row should state the calculation, the interval or point, and the graphical consequence. This makes the final sketch a summary of evidence instead of an artistic guess.
Read this graph as text
A derivative sign chart turns algebra into a graph story. Critical numbers divide the domain into intervals. Testing the sign of f' on each interval determines whether the original function rises or falls and classifies sign changes. A positive derivative means the original function increases; a negative derivative means it decreases. At x=-2 , the sign changes from positive to negative, so f has a local maximum. At x=1 , the sign changes from negative to positive, so f has a local minimum.
Every relationship in a derivative sign chart turns algebra into a graph story is identified with written labels plus distinct solid, dashed, dotted, double, marker, or pattern cues; color is never the only carrier of meaning.
Why it matters: The chart should bridge symbolic factor signs and graph behavior. It is a reusable component for increasing/decreasing intervals, the first derivative test, and optimization verification.
Critical numbers divide the domain into intervals. Testing the sign of f' on each interval determines whether the original function rises or falls and classifies sign changes.
A turning point is detected by a sign change
At a local maximum, the function changes from increasing to decreasing, so changes from positive to negative. At a local minimum, changes from negative to positive. If the sign does not change, the critical point is not a local extremum.
This test describes behavior on both sides and works even when is undefined at the critical point, provided the original function is defined there.
The first derivative test classifies a critical point by observing what the function does on either side. A change from increasing to decreasing produces a local maximum; decreasing to increasing produces a local minimum; no sign change produces neither.
This test handles cases where is zero or undefined, so it is often more robust than the second derivative test. It also explains the result through actual motion of the graph rather than a memorized sign table.
First Derivative Test
At a critical number :
• if changes from positive to negative, has a local maximum; • if changes from negative to positive, has a local minimum; • if does not change sign, there is no local extremum.
For the previous cubic, changes at , so has a local maximum there. It changes at , so has a local minimum there.
A critical point that is not an extremum
Classify the critical point of at .
Worked solution
Write a real attempt before opening the supplied answer.
Solving finds candidates. It does not classify them. A zero derivative may be a maximum, minimum, flat inflection point, or part of a constant interval.
Peak concentration after a dose
Suppose
models drug concentration for . Then
The exponential factor is positive, so the derivative changes sign when , at . Concentration increases before hours and decreases after, so the first derivative test identifies a peak at .
app-drug-peak-01For , at what time is concentration maximal?
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Factor and locate its sign change.
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